296 BKLL SYSTEM Tl'.CIIMCAL JOURNAL 



and 



jR- = Zu-^-ik = ^' = a constant. 



It will be noted from these formulas that the transfer constant and 

 both image impedances of any "constant ^" wave-filter are functions 

 of frequency only through the variables Ut + iVk, or the equivalent 

 (su-/27?)- which is a function of su. (It would also hav^e been possible 

 to use So/.: instead of Zu--) When no dissipation in the elements is 

 assumed, Su = ru- + ^^u becomes Zi/, = ixv;, a pure reactance, since 

 then Vik — 0; also Fa = 0. Under these ideal conditions we know that 

 X\k always has a positive slope with frequency,^ and when the Xik of a 

 multiple band wave-filter is plotted against frequency it is made up of 

 negative branches from .vu = — oc to and positive branches from X\k 

 = to + 3c which lie alternately in succession along the frequency 

 scale. These branches are defined to correspond with the sign of .tu. 

 The value of Lh is always negative and ranges continuously with 

 frequency between the values Uk = and — cc , once for each branch 

 of Xik. We know also that in a negative branch there is a transmitting 

 band at frequencies corresponding to values from .vn = — 2R to 0, and 

 thus from Ui, = — 1 to 0. In a positive branch there is a transmitting 

 band from xu = to + 2R, thus from Uh = to — 1. A low pass 

 band is associated with a positive branch which begins at zero fre- 

 quency while a high pass band is associated with a negative branch 

 ending at infinite frequency. An internal transmitting band, on the 

 other hand, has this association with a pair of branches, a negative 

 followed on the frequency scale by a positive branch, and in reality 

 consists of two bands which are confluent at .Vu- = 0, i.e., Uk — 0, 

 where the two branches join. Such a confluent band is formed by the 

 junction of two bands which occur separately in a wave-filter of higher 

 class than this "constant k" wave-filter but with the same configura- 

 tion of elements. 



Since all negative branches are similar, as well as all positive 

 branches, an approximate representation of the frequency charac- 

 teristics of any "constant ^" wave-filter can be constructed from the 

 characteristics which belong to each of these two kinds of branches. 

 It is necessary to consider both a negative branch and a positive 

 branch since the characteristics of one branch dift'er in their variations 

 with frequency from those of the other. Dift'erences would naturalh' 

 be expected from the fact that in formulas (6) which hold for both 

 branches the variable Uk varies with increasing frequency from 

 Uk = — =« to in a negati\'e branch and from f/i = to — ^ in a 



* See page 5 of paper in footnote 1. 



